Finally, let us recall that the sequence \({\bf a}\)  with values in \(\mathbb {Z}\) -module \(R\)  is \(k\) -regular if there exist finitely many sequences \(\mathbf {a}_{i} = (a_i(n))_{n\in \mathbb {N}}\)  with values in \(R\)  such that each sequence in \(K_k({\bf a})\)  is a \(\mathbb {Z}\) -linear combination of the \(\bf {a}_{i}\) . In other words, the \(\mathbb {Z}\) -submodule generated by the \(k\) -kernel \(K_k({\bf a})\) . In particular, \(k\) -automatic sequences are precisely \(k\) -regular sequences taking finitely many values. The class of \(k\) -regular sequences with values in a ring \(R\)  has a ring structure itself. A good introduction to the topic of regular sequences are the papers of Allouche and Shallit [1]}, [2]}.
